## mathslover 2 years ago Prove that if the 3 altitudes of a triangle are equal then the triangle is equilateral. See attachment

1. mathslover

I tried to do this and have got so far that AB = AC

2. mathslover

@ParthKohli

3. mathslover

@experimentX

4. mathslover

I took following triangles :ABF and AEC by AAS I got both of these triangles congruent and therefore by CPCT AB = AC

5. mathslover

(CPCT = Congruent Parts of Congruent Triangles are equal)

6. joemath314159

That idea should work to get that AB=BC as well.

7. mathslover

So far , we have got ABC as isosceles. Now, I took triangle ABC and AFC both of these triangles are congruent by RHS BF = FC ( by CPCT ) therefore F is mid point of BC an AF is median and altitude also Therefore , this is the property of an equilateral triangle , so ABC is an equi. triangle

8. mathslover

Is this correct?

9. mathslover

You mean to take ABE and BEC triangles @joemath314159 ?

10. joemath314159

yep. and do exactly what you did with the other triangles again.

11. mathslover

I meant this : ADC and ABD in the above comment.

12. mathslover

OK wait..

13. mathslover

I think I had tried that earlier but I didnt' get any favorable result , wait let me do it again

14. joemath314159

there are many combinations you could take. The pair I see is triangles BEC and BDA.

15. mathslover

Oh yes that will work ... BEC and BDA .. this gives some output to me.

16. mathslover

Thanks a lot @joemath314159 for your help .. Might gonna work in more concentration for me..